How to Operate on Functions

By Krystle Rose Forseth, Christopher Burger, Michelle Rose Gilman, and Deborah Rumsey

You can operate on (sometimes called combining) functions pretty easily. You may have to perform operations on functions — such as addition, subtraction, multiplication, and division — as well as break down function compositions.

Adding and subtracting functions

In the following examples,

When asked to add functions, you simply combine like terms, if the functions have any. For example, (f + g)(x) is asking you to add the f (x) and the g (x) functions:

remains because it has no like terms; and 1 and –10 add to –9.

But what do you do if you’re asked to add (g + h)(x)? You’d get

You have no like terms to add, so you can’t simplify the answer any further. You’re done!

When asked to subtract functions, you distribute the negative sign throughout the second function, using the distributive property, and then treat the process like an addition problem:

Multiplying and dividing functions

Multiplying and dividing functions is a similar concept to adding and subtracting them. When multiplying functions, you use the distributive property over and over and then add the like terms to simplify. Dividing functions is trickier, however. You’ll tackle multiplication first and save the trickster division for last. Here’s the setup for multiplying f (x) and g (x):

Follow these steps to multiply these functions:

Distribute each term of the polynomial on the left to each term of the polynomial on the right.

You start with

You end up with
Combine like terms to get the final answer to the multiplication.

This simple step gives you

Operations that call for division of functions may involve factoring to cancel out terms and simplify the fraction. If you’re asked to divide g (x) by f (x), though, you’d write

Because neither the denominator nor the numerator factor, the new, combined function is simplified and you’re done.

You may be asked to find a specific value of a combined function. For example, (f + h)(1) asks you to put the value of 1 into the combined function (f + h)(x) =